High-order accurate adaptive numerical methods for fluid mechanics using unstructured meshes

为了使复杂流动的计算流体力学(CFD)模拟结果在工程设计环境中有用，模拟软件(求解器)必须高效而稳健地产生准确的解，同时最大限度地减少建立模拟所需的人力时间。为了解决这些问题，我的研究小组开发了非结构网格的高阶精确解技术。我们的结果和其他人的结果一致表明，使用高阶方法可以更快地获得相同精度的流动解。我的团队的长期目标是将高阶方法的好处从研究层面带入工业可用性。在短期内，我们将强调高阶方法在构型空气动力学、气动优化和流固耦合中的应用。为了使这些应用在高阶方法的背景下实现，我们将扩展现有的二阶技术或开发新的粘性网格生成、网格自适应和求解伴随问题的技术。他说，高阶离散格式在粘性网格生成中的关键问题是曲面的准确表示，包括创建带曲面的网格单元以避免壁上的倒置单元。他说，我们已经在曲线边界网格划分方面取得了重大进展，并期待在粘性网格划分工作中进一步发展。高阶网格自适应程序将需要开发新的误差指示器，因为二阶自适应方法常用的二阶导数已经被高阶求解器捕获。我们将研究使用高阶导数作为方向误差指示器，并执行网格自适应以有效控制误差。我的团队早期在高阶方法优化方面的努力因难以计算足够精确的梯度而受阻。共轭问题的高阶解应该提供所需的梯度，并在计算解的误差界限方面也有应用。

For the results of computational fluid dynamics (CFD) simulations of complex flows to be useful in an engineering design context, the simulation software (solver) must produce accurate solutions efficiently and robustly while minimizing the human time to set up a simulation.      To address these issues, my research group has developed high-order accurate solution techniques for unstructured meshes.  Our results and those of others consistently show that a flow solution of the same accuracy can be obtained more quickly using high-order methods.  My group's long-term goal is to bring the benefits of high-order methods from the research level to industrial usability.      In the near term, we will emphasize application of high-order methods to configuration aerodynamics, aerodynamic optimization, and fluids-structures interaction.  To enable these applications in the context of high-order methods, we will extend existing second-order techniques or develop new techniques for viscous mesh generation, mesh adaptation, and solution of adjoint problems.      The key issue in viscous mesh generation for high-order discretization schemes is the accurate representation of curved surfaces, including creation of mesh cells with curved sides to avoid inverted cells at the wall.  We have already made significant progress on curved-boundary meshing, and expect to build on that in the viscous meshing work. High-order mesh adaptation procedures will require the development of new error indicators, as the second derivatives commonly used by second-order adaptive methods are already captured by a high-order solver.  We will investigate the use of higher-order derivatives as directional error indicators, and perform mesh adaptation to control error effectively.  My group's early efforts in optimization for high-order methods were stymied by difficulties in computing sufficiently accurate gradients.  High-order solution of adjoint problems should provide the required gradients, and also have application in computing solution error bounds.